NOTE: This
page is a continuation of the notes and worksheets for topic 9.7 Astrophysics.
Four separate pages were used for this topic because of the large volume of
material in the topic. This will keep download time within acceptable
limits.

NOTE:
Numbers appearing in parentheses at the end of sentences or paragraphs refer to
the references provided in the Bibliography at the end of these notes.

THE
RED GIANT STAGE IN THE LIFE OF A STAR

When the hydrogen has been
exhausted in the core of a main sequence star, hydrogen burning ceases.This leaves a core consisting almost entirely of helium, surrounded by a
shell through which hydrogen burning works its way outward in the star.

When the hydrogen burning stops
in the core, the temperature there decreases causing a corresponding decrease in
pressure.The core contracts under
the weight of the outer layers of the star.As the core contracts it becomes hotter and the heat flows outwards
warming the gases around the core and increasing the rate of shell hydrogen
burning.Helium produced by
these reactions falls back into the core, which continues to contract and heat
up as it gains mass.Over the
course of hundreds of millions of years, the core of a one solar mass star
compresses to about one-third of its original radius, while the core temperature
increases from about 15 million K to about 100 million K.(3)

While the core contracts, the
hydrogen burning continues to move outwards causing an increase in the star’s
luminosity and an increase in the star’s internal pressure.This increase in pressure makes the entire star expand to many times its
original radius.This massive
expansion of the star’s outer layers causes the star’s surface temperature
to decrease.Once the surface
temperature has reached about 3500 K, the gases glow with a reddish hue, in
accordance with Wien’s Law.The
star is then called a Red Giant star.Red Giant stars are stars that have finished their time on the main
sequence and have evolved into a new stage of existence.(3)

Examine the evolutionary
track for a one solar mass star shown on H-R diagram (a) below.Make sure you can explain the shape of this track in terms of the
physical processes described above.

In a moderately low mass red
giant, which the Sun will be in another 5 billion years or so, the dense helium
core is about twice the size of the Earth and the star’s bloated surface has a
diameter of about 1AU.

When a star first becomes a red
giant, the temperature of the contracted helium core is still too low for helium
fusion to commence.In time, as the
hydrogen burning shell continues to add mass to the helium core, the core
contracts even more, further increasing the core temperature.When the core temperature reaches about 100 million K, the fusion of
helium (helium burning) begins there.This
process, also called the triple alpha process, converts helium to carbon
and oxygen.In a high-mass red
giant (mass > 2 to 3 solar masses), with a hotter core, helium burning begins
gradually whereas in a low-mass red giant (mass < 2 to 3 solar masses), it
begins very suddenly, in a process called the helium flash.

Note that for red giants with
masses less than about 0.5 solar masses, the core will never reach the
temperature required for helium fusion and the core and shell both contract and
become hotter until the star has become a white dwarf (9).
The reactions involved in the triple alpha process are as follows:

Clearly, in step 1, two helium
nuclei combine to form a very unstable isotope of beryllium.In step 2, a third helium nucleus collides with the beryllium
nucleus to form a stable isotope of carbon.A gamma ray photon is emitted in this process.Note that the name “triple alpha process” arises from the
common name for the helium nucleus – the “alpha particle”.

Some of the carbon produced in
the triple alpha process can fuse with another helium nucleus to produce a
stable isotope of oxygen, as shown below:

The triple alpha process
produces only 10% of the energy per kilogram of fuel compared with hydrogen
burning (12).A mature red giant
burns helium in its core for about 20% as long as the time spent on the main
sequence (3).So, for example, in
the distant future our Sun will burn helium for about 2 billion years.

Once the helium burning process
has begun in the core of a red giant, the star undergoes further changes.The star’s superheated core expands like an ideal gas.Around the expanding core, temperatures fall, so the hydrogen burning
shell reduces its energy output and the star’s luminosity falls.This allows the star’s outer layers to contract and heat up.Thus, after the onset of helium burning, a red giant is less luminous,
is hotter at the surface and is smaller than it was before helium burning
started.

Study the evolutionary track for
a one solar mass star shown on H-R diagram (b) below.In accordance with the physics described above, the track
moves toward lower luminosity and higher surface temperature, following a track
called the horizontal branch.Stars
along the horizontal branch have helium-burning cores surrounded by
hydrogen-burning shells.

The changes that occur in red
giants after the onset of helium burning cause some red giants to become
unstable.For instance, the
hydrogen burning shell may become sufficiently unstable to cause the star to pulsate
as a periodic variable, driven by the changed radiation pressure within
(12).

AFTER THE CORE
HELIUM-BURNING STAGE

Eventually all the helium in a
star’s core is exhausted and core helium-burning ceases.What happens next depends critically upon the ZAMS mass of
the star.(1)

STARS
LESS THAN 4 SOLAR MASSES

Let us deal first with low
mass stars of less than 4 solar masses.The core consists of carbon and oxygen.Around the core is a shell of helium.Surrounding the helium shell is the now dormant hydrogen shell.Without thermonuclear reactions to maintain the core’s internal
pressure, the core contracts, releasing heat into the helium shell.Fusion of helium begins in this shell.(3)

The energy released from the
shell helium burning causes the outer layers of the star to expand again.Luminosity increases, surface temperature decreases and the star enters a
second red giant phase.These stars
are called asymptotic giant branch stars (AGB stars) and their
evolutionary tracks follow the asymptotic giant branch on the H-R diagram.See below.Stars of less
than 4 solar masses in the AGB phase are nearing the final stage in their lives.We will complete their story soon.

Just as an aside, it is
interesting to consider why stars of less than 4 solar masses cannot simply
start fusing the carbon and oxygen in their cores to produce more energy.The reason is that electron degeneracy pressure stops the cores of
such stars from contracting sufficiently to produce the required temperatures
for the fusion of carbon and oxygen.The
Pauli Exclusion Principle tells us that no two electrons can occupy the
same quantum state at the same time.This
effectively places a limit on the degree to which a substance can be compressed
because eventually the electrons will be so close together that any further
compression would violate the exclusion principle.(3)

Let us now consider what happens
to a star of 4 solar masses or more after core helium burning ceases.For such a star, the core is sufficiently massive to continue
contracting, increasing the core temperature.When the core temperature reaches 600 million K, carbon burning
commences.This fusion process
produces oxygen, neon, sodium and magnesium.(3)

For stars with a ZAMS mass of 8
solar masses or greater, the cessation of carbon burning results in further
contraction and heating of the core.At
a core temperature of around 1 billion K neon burning begins, which uses
up the neon accumulated from the carbon burning and increases the amounts of
oxygen and magnesium in the star’s core.

Once the neon burning has
finished, the core will contract again and oxygen burning will commence
at around 1.5 billion K.This
reaction produces sulfur.

When oxygen burning is complete,
the core will contract again and silicon burning commences at around 2.7
billion K.Silicon fusion produces
several nuclei from sulfur to iron.(3)

Each new stage of core burning
generates a new shell of material around the core.After several such stages, the internal structure of a very massive star
(eg a 25 solar mass star) resembles that of an onion as shown in the diagram
below.Note that iron is the
final element that is produced by the fusion reactions occurring inside the core
of a massive star.The
fusion of iron or any element heavier than iron consumes energy rather than
releasing it.(3)

Between each new stage of core
burning comes a period of shell burning and a new red giant phase for the star.This means that the evolutionary tracks of high mass stars go through a
series of back-and-forth gyrations on the H-R diagram.See the diagram below showing the evolutionary track for a star of about 10
solar masses.(3)

The energy released by the
processes described above causes the star’s outer layers to expand greatly.The result is a Supergiant star.The largest supergiants are a thousand times larger than our present-day
Sun, with diameters as large as the orbit of Jupiter around the Sun.(3)

Betelgeuse and Rigel in the
constellation of Orion and Antares in the constellation of Scorpius are easily
observable examples of supergiant stars.Spring/summer
is best for observing Orion in Australia and autumn/winter for Scorpius.

The diagram above has been
adapted from Figure 22-13 on p.550 of Ref.3.It shows a high-mass star that has become a supergiant.Its diameter is almost as large as the orbit of Jupiter
around the Sun.The star’s energy
comes from six concentric burning shells, all contained within a volume roughly
the size of Earth.No thermonuclear
reactions occur in the iron core, since fusion reactions that involve iron
absorb energy rather than releasing it

NUCLEOSYNTHESIS

Nucleosynthesis
is the process of creating elements by nuclear reactions (5).The Syllabus requires that you are able to “discuss
the synthesis of elements in stars by fusion”.In the
“Cosmic Engine” topic in the Preliminary
Course we saw that
hydrogen, helium and lithium were created in the Big Bang.We have seen in this topic that hydrogen is fused into helium in main
sequence stars, that helium is fused into carbon and oxygen in red giant stars
and that all elements up to and including iron can be produced by fusion
reactions in the cores of supergiants.Theoretically,
you already have sufficient information to successfully answer this Syllabus
point, since the remaining elements in the Periodic Table are not produced by
fusion reactions.

However, for completeness we will mention the two main
processes that are believed to be responsible for the production of the elements
heavier than iron.The first of
these is the slow neutron capture reaction that occurs in the shells of AGB
stars.The s-process, as
the reaction is called, involves the capture of neutrons by existing nuclei (eg
Fe-56) to form heavier ones.Unstable
nuclei formed in this way then undergo the beta-decay process to produce new
elements.This process can form
elements up to and including lead.Once
s-process elements are formed, the AGB star conveniently convects these to the
surface, where they may be released either in a stellar wind or in a subsequent
supernova explosion.The second
process is the rapid neutron capture reaction, also called the r-process.This occurs during type II supernova events and builds on iron to produce
all of the heavier elements found in the periodic table.(1 & 15)

Much research
continues into the question of how the elements from iron to uranium were made.The rapid proton capture reaction which is believed to be the cause of
Nova explosions and X-ray bursts is being investigated as a possible source of
heavier elements (15).The
possibility that fusion reactions involving iron and progressively heavier
elements may be fueled by the huge energy output of a supernova explosion is
also being examined (3).

DETERMINATION OF
THE AGE OF A STAR CLUSTER

A group of stars formed
together from the same giant molecular cloud and held together by gravity is
called a cluster (5).There are
two main types of cluster – Open Clusters and Globular Clusters.These are distinguished on the basis of appearance, age, size and
position in the galaxy.Stars in a cluster are of approximately the same age.(9)

Open clusters contain from
a few dozen stars up to a few hundred together with dust and gas from which new
stars may be forming.The stars are
far enough apart to be resolved by the naked eye or a telescope.Open clusters occur in the galactic plane and are therefore sometimes
referred to as Galactic Clusters.Open
clusters contain some hot, massive O and B class stars.Since such stars have short lifetimes, open clusters are relatively
young.Good examples of easy to
find open clusters in the southern hemisphere include the Jewel Box, the Scorpio
Clusters M6 and M7 and the Pleiades.(9)

Globular clusters contain
from several thousand stars up to several million arranged in spherical-shaped
geometries.They contain relatively
little dust or gas and no high mass main sequence stars.In fact the stars in them are as old as the galaxy itself.The stars are also very close together, astronomically speaking, and so
most stars in the central region of a globular cluster cannot be resolved by
earth-based telescopes.Globular
clusters are located in the galactic halo region above and below the galactic
plane.They are “left-overs”
from the formation of the galaxy.There
are about 150 known globular clusters in our galaxy and possibly up to about 200
total (14).The brightest example
is w-Centauri,
clearly visible with binoculars from Sydney’s latitude all year round.(9 & 10)

The age of a cluster can be
determined by plotting an H-R diagram of the cluster.In an H-R diagram for a very young cluster, all the stars lie on or near
the main sequence.As time goes by the high-mass, high-luminosity stars are the
first to evolve away from the main sequence, as they become red giants.Over the years the main sequence gets shorter and shorter.(1 & 3)

The age of a cluster can be
determined from the turnoff point, which is the top of the surviving portion of
the main sequence on the cluster’s H-R diagram.The stars at the turnoff point are just completing their
hydrogen-burning phase, so their main sequence lifetime is equal to the age of
the cluster.Stellar modelling
based on the nuclear processes occurring in stars has enabled main sequence
lifetimes to be associated with each particular turnoff point.Thus, the age of the cluster can be determined.(1 & 3)

Age determinations have been made
for many clusters.The oldest
clusters are of course the globular clusters and seem to be almost as old as the
universe itself – 12 to 15 billion years.Open clusters are much younger.The
Pleiades cluster, for example, is estimated to be about 100 million years old.(12)

Clearly, the accuracy and
reliability of the turnoff point determination is paramount.The position of the turnoff point can best be determined by overlaying a
theoretical ZAMS (zero-age main sequence) plot drawn to the same scale on
top of the H-R diagram for the cluster under study.With the appropriate x-axis scales aligned, the ZAMS plot can then be
moved vertically to obtain the best match of the theoretical ZAMS with the
observed main sequence on the H-R diagram of the cluster.The turnoff point, where stars are starting to move away from the main
sequence, is then usually easy to define.

Be aware that once the turnoff
point is identified, there are a few different methods available for determining
the age of the cluster depending on what information you have available.

Let us look at an example of the
process.Study the colour-magnitude
diagram (equivalent to an H-R diagram) for the M55 globular cluster,
shown below.

Each dot in this diagram
represents the apparent magnitude (measured through a V filter) and surface
temperature (as measured by the colour index, B – V, adjusted for interstellar
reddening) of a star in the cluster.Since
all of the stars in a cluster are essentially at the same distance from Earth
(about 6000 pc in this case), the apparent magnitude (a measure of apparent
brightness) is a direct measure of luminosity.(3)

As described above a theoretical
ZAMS H-R diagram would be fitted to the main sequence on the H-R diagram of the
cluster.This highlights the
position of the turnoff point, making it easier to identify.Once this has been done, one could use the B-V value
at the turnoff point and some mathematical manipulation to obtain an age for the
cluster.Alternatively, one could
use the luminosity value at the turnoff point, to calculate the mass of the
stars at that point and then use an equation connecting star mass with main
sequence lifetime to calculate the age of the cluster.

In the example of the M55
globular cluster, the age of the cluster works out to be between 13 and
15 billion years depending on the method used.

Note that in our study of the age
determination of clusters we have stated that the age of the cluster is equal to
the main sequence lifetime of stars at the turnoff point.Some could argue that the age of the cluster must be measured from the
time of the initial collapse of the molecular cloud (1).Strictly speaking this is true.However,
the time taken for the pre-main sequence evolution of the stars at the turnoff
point is only a very small fraction of the time spent by those stars on the main
sequence (3).Therefore, it is
reasonable to approximate by saying that the age of the cluster is equivalent to
the main sequence lifetime of stars at the turnoff point.

Extension – Non-Examinable:

Just for the
record, horizontal branch stars are post-helium-flash, low-mass stars
with luminosities of about 50 times that of the Sun and in which there is both
core helium burning and shell hydrogen burning.These stars will eventually move back toward the red giant region as
their fuel is exhausted.

Also for
those who are interested, for main sequence stars, the relationships between the
luminosity, mass and time on the main sequence are as
follows:

Luminosity
is directly related to mass:

Lifetime on
the main sequence, t,
depends critically on both mass and luminosity:

Joining
these two equations together we have:

These
equations can be used to calculate estimates of main sequence lifetimes.

STELLAR
DEATH

“Out, out brief candle! Life’s but a walking
shadow, a poor player that struts and frets his hour upon the stage and then is
heard no more.”Shakespeare – Macbeth Act 5 Scene 5.

All material things come to an
end.As powerful and as majestic as
all stars are, eventually they grow old and die.Eventually all stars reach a stage where the material in their cores
cannot undergo further fusion.This
could be because the core cannot contract any further to reach the temperature
required for the next possible set of fusion reactions or it could be that the
core consists of iron, which will not fuse to produce energy.Around these dormant cores, fusion continues in the various shells that
each star has developed during its lifetime.In their death throes all stars shed their shells into space and undergo
core collapse by gravity.

STARS
OF 8 SOLAR MASSES OR LESS

Current research indicates that
all stars with ZAMS masses of 8 solar masses or less shed a large portion
of their mass during the AGB phase of their evolution.During this phase, alternating ignitions of hydrogen and helium burning
shells produce the energy in the star.As
each new helium-shell-burning episode begins bursts of energy known as thermal
pulses spread outwards through the star blowing mass into space.A 1 solar mass star loses about 40% of its mass.The more massive the star, the higher the proportion of its initial mass
that is lost.(1 & 3)

During these thermal pulses, the
outer layers of the star can separate completely, exposing the hot core.Ultraviolet radiation from the exposed core ionizes and excites the
expanding shell of ejected gases.These
gases therefore glow and emit visible light, producing a planetary nebula.Over time, perhaps no more than 50 000 years, the planetary nebula
disperses, cools and fades from view.Note
that the name “planetary nebula” is really a misnomer.It comes from the fact that through a small telescope the expanding shell
of glowing gas can look like a planet.(1
& 3)

No more nuclear reactions take
place in the exposed core.If it
has not already done so, the core collapses under gravity and provided its
mass is not greater than 1.4 solar masses, it becomes a degenerate
(non-contracting), dense sphere about the same size as the Earth.The star is now called a White Dwarf.The interior of a white dwarf consists mainly of carbon and
oxygen atoms floating in a sea of degenerate electrons.The White Dwarf gives off thermal radiation, which causes it to
glow.As it cools, it remains the
same size, since the degenerate electron pressure does not depend on temperature
but becomes less luminous, eventually fading into obscurity as a black dwarf.(1 & 3)

Note that the 1.4 solar mass
limit to the size of a white dwarf is called the Chandrasekhar limit,
after the Indian-American Physicist Subrahmanyan Chandrasekhar, who did
pioneering research on white dwarfs.The
Chandrasekhar limit is the maximum possible mass of a degenerate star, above
which it will be unable to support itself against the inward pull of its own
gravity (5).

It is worth considering at this
stage what the evolutionary tracks of stars look like as they become white
dwarfs.Starting from the top of
the AGB, the track moves horizontally from right to left across the H-R diagram
as the temperature increases.In
some cases loops appear on the diagram corresponding to thermal pulses.As the planetary nebula and core cool, luminosity decreases and the track
turns down towards the lower right hand corner of the H-R diagram.See the excellent H-R diagram below for a 1 solar mass star, taken
from the Sloan Digital Sky Survey website at:

Stars with ZAMS masses of greater
than 8 solar masses experience a different death. Stars with more than
about 8 solar masses explode as type II supernovae after a lifetime of
only a few million years and become neutron stars or black holes.(3)

Such massive stars still
experience the large mass loss due to thermal pulses and the core collapse that
less massive stars experience.The
difference, however, is that with these more massive stars the core is greater
than 1.4 solar masses when core collapse begins.So, as the core collapses under its own gravity the electron degeneracy
pressure is unable to stop the collapse.In
a fraction of a second, the core becomes so dense that electrons and protons
are forced to combine to form neutrons.This process releases neutrinos that carry large amounts of energy out of
the core, cooling it dramatically.If
the mass of the core is less than about 3 solar masses, the neutron
degeneracy pressure in the core brings the collapse to a sudden stop.At this stage the density of matter in the core is about 1017
kg/m3 – the density of nuclear matter.(3 & 5)

Due to the sudden stop of the
collapse, the innermost portion of the core bounces back and expands.This causes a powerful pressure wave to spread outwards from the core.At the same time, the dramatic cooling of the core caused by the escaping
neutrinos reduces the pressure in the regions surrounding the core and causes
the matter in these regions to fall inwards at speeds up to 15% the speed of
light.When the inward-falling
material hits the rigid iron core, it meets the outward-moving pressure wave.The result is an outward–moving shock wave that totally disrupts the
star.The energy released by
this shock wave is of the order of 1046 joules – a hundred times
more energy than the Sun has emitted in its whole lifetime.(3)

The energy spreading throughout
the star drives the reactions that produce all the elements heavier than iron.The shock wave blows most of the matter that made up the star out into
space as a cloud of expanding gas and dust.This matter will one day seed the formation of new stars.(3)

During this explosion, the
luminosity increases up to several billion times as the core is partly exposed
and the star appears brighter than an entire galaxy.This increased brightness lasts for several days and fades
away over a period of months.This
explosive event is called a supernova (plural - supernovae) and the
dense, collapsed stellar core left behind is referred to as the supernova
remnant.(1 & 3)

As an aside, it is worth
mentioning that the event described above is a type II supernova,
resulting from the death of a massive star.SN1987A is an example of a type II supernova that was observed in
the Large Magellanic Cloud in 1987.The
star that exploded in this case is believed to have been a blue supergiant star
with a ZAMS mass of about 20 solar masses.Other types of supernova exist. Type Ia supernovae for example
result from the catastrophic explosion of a white dwarf star in a close binary
system.It is caused by runaway
carbon fusion in the white dwarf triggered by mass transfer from its companion
red giant star.(3)

The nature of the supernova
remnant depends critically on the mass of the core of theprogenitor
star – the star that exploded.For
stellar cores greater than 1.4 solar masses but less than about 3 solar
masses a neutron star results.As
described above, the matter within such a core has been crushed to nuclear
densities and exists as neutrons.The
neutron star is protected against further gravitational collapse by the
degenerate pressure of neutrons.A
typical neutron star with a mass a little greater than the Sun’s would have a
diameter of only 30 km and a density such that the mass of the entire human race
would occupy the volume of a sugar cube!Neutron
stars also possess huge magnetic fields, of the order of 1 x 108 T.Compare this to the magnetic field at the surface of the Sun – around 10-4
T.(3 & 5)

All neutron stars spin rapidly due
to the conservation of angular momentum of the original star.As a star shrinks down to become a neutron star, its rotation must speed
up.Charged particles that have
been accelerated near the neutron star’s magnetic poles produce two oppositely
directed beams of radiation that emanate from the magnetic poles.As the star rotates, these beams sweep through space.If the Earth happens to lie in the path of the beams, we detect radiation
that appears to pulse on and off.Such a neutron star is called a pulsar.The radiation it emits can be of varying kinds – radio, optical, X-ray
and gamma.See diagram below.The first pulsar was discovered by Jocelyn Bell at Cambridge University
in 1967.The Crab Nebula
is the remnant of a supernova explosion in the constellation Taurus first
observed in 1054. The stellar corpse at its centre is a pulsar.(3 & 12)

If the core of the exploding
star is greater than about 3 solar masses, nothing can stop its gravitational
collapse to a black hole.A
collapsing star becomes a black hole when its radius has shrunk to a critical
size, known as the Schwarzschild radius, at which gravity is so strong
that not even light can escape from the surface of the star.The surface having this critical radius is called the event horizon
and marks the boundary inside which all information is trapped.Hence, events within a black hole cannot be observed from outside the
event horizon.Theory indicates
that both space and time become distorted inside the event horizon and that an
object collapses to a single point, a singularity, at the centre of a
black hole.(3 & 5)

Do black holes really exist?Today, there is compelling astrophysical evidence that they do.Many examples of possible black holes have been found.Cygnus X-1 is a strong source of X-rays in our own galaxy that is
believed to be a black hole.Supermassive
black holes are believed to lie at the centres of many galaxies including our
own.(3)

REVISION
TEST No.1

(a)The accurate measurement of distance is extremely important in astronomy.Careful measurement of a celestial object’s position in the sky may be
used to determine its distance.

(i)A certain star, has a parallax angle of 0.258 arcsec.What is the distance of the star from Earth? (1 Mark)

(ii)Define the terms “sensitivity” and “resolution” and explain why
it is desirable for telescopes to have a large diameter objective lens or mirror
in terms of both sensitivity and resolution. (3 Marks)

(i)A visual binary star system consists of two stars, one of which orbits
the other in a near circular orbit, once every five years at an average
distance of 1.5 x 1012 m.What
is the sum of the masses of the two stars in kilograms? (3 Marks)

An astronomer observes a Type I Cepheid variable in a distant galaxy.The Cepheid has a period of 3 days and an average apparent
magnitude of +12.The astronomer
claims that this data indicates that the galaxy is approximately 12 600 parsecs
away.

Use your knowledge of how the period-luminosity relationship can be used to
calculate distance to justify the astronomer’s claim. (3 Marks)

(c)Spectroscopy is an essential tool for astronomers and provides a wealth
of information.

(i)Surface temperature, rotational velocity and chemical composition are
three factors that can be determined from the spectrum of a star.For each factor identify the feature of the spectrum that allows the
factor to be determined. (3 Marks)

MARKING
GUIDELINES - Revision Test No.1

(ii)1 mark for correct definition of “sensitivity”.
1 mark for correct definition of “resolution”.
1 mark for correct explanation of reason for large objective.

Notes for (a) (ii):
For optical telescopes, sensitivity refers to the light-gathering power of the
telescope & is directly proportional to the square of the diameter of the
objective.The resolution (angular
or optical) of a telescope is the minimum angular separation between two equal
point sources such that they can be just barely distinguished as separate
sources.

qmin = 1.22l /D in
radians or qmin = 2.5 x 105l/D in arcseconds.

The smaller the angle, the finer the details that can be seen and the sharper
the image.Clearly, the larger the
diameter, D, of the objective, the more sensitive the
telescope (ie the larger D2) & the better
the resolution (ie the smaller qmin).

(b)

(i)1 mark for use of correct formula.
1 mark for correct period in seconds.
1 mark for correct answer of 8.02 x 1031 kg.

(ii)1 mark for recognizing Algol as an eclipsing binary.
1 mark for recognizing that at A & C the brighter member is moving behind
the duller member.
1 mark for recognizing that at B the brighter member is moving across the
duller.

(iii)1 mark for determining that the absolute magnitude, M = -
3.5.
1 mark for use of correct formula (M = m – 5log[d/10]).
1 mark for SHOWING the whole calculation process and arriving at the correct
answer (12 589 pc = 12 600 pc approx).Students MUST show sufficient working to convince the marker that they
know how to use the formula correctly.

(c)

(i)1 mark for stating that surface temperature is calculated from the
wavelength of maximum emission in the spectrum of the star.
1 mark for stating that rotational velocity is determined from the Doppler
broadening of lines in the spectrum of the star.
1 mark for stating that chemical composition can be deduced by identifying the
groups of spectral lines characteristic of individual elements.

(ii)Answers scoring 2 marks should accurately describe some form of
spectrograph.eg a Grating
Spectrograph - an optical device that uses a diffraction grating to break up the
light from a source into a spectrum.A
collimator ensures that light rays striking the grating are parallel.A corrector lens and mirror then focus the spectrum onto a CCD
(charge-coupled device), which records the image and usually feeds the data
straight to a computer for analysis.
Answers scoring 1 mark would partially describe some form of spectrograph.
Note that diagrams are not essential here but could be drawn.